Generating trees for permutations avoiding generalized patterns
Sergi Elizalde

TL;DR
This paper constructs generating trees for classes of permutations avoiding generalized patterns, enabling the derivation of functional equations and enumeration results, including refinements and new findings.
Contribution
It introduces a novel tree-based method to enumerate permutations avoiding generalized patterns of length 3 and 4, solving associated functional equations.
Findings
Derived functional equations for permutation classes
Solved equations using kernel method and Bousquet-Mélou's ideas
Refined and discovered new enumerative results
Abstract
We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation, which allows us to incorporate the adjacency condition about some entries in an occurrence of a generalized pattern. We use these trees to find functional equations for the generating functions enumerating these classes of permutations with respect to different parameters. In several cases we solve them using the kernel method and some ideas of Bousquet-M\'elou. We obtain refinements of known enumerative results and find new ones.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Mathematical Dynamics and Fractals
